For real applications, the choice of the covariance model is case-specific. Understanding the spatio-temporal variability within and across variables is essential in choosing a realistic covariance model. In addition, with the development of remote sensing and in situ measurement techniques, and with powerful computing facilities that enable better physical model simulations, large data sets of unprecedented size are collected.
It is often challenging and slow to perform the model inference because space-time covariance models typically involve many parameters that need to be estimated from data. Specific constraints on the proposed covariance structure can reduce the model complexity and accelerate the inference procedure.
These considerations motivate researchers to study simplified covariance structures and propose tools to visualize and assess them.
[1]} proposed several approaches to build univariate nonseparable spatio-temporal covariances with separable covariances as special cases.
[2]} investigated both the univariate separability and symmetry covariance structures.
Graphical evidence for these univariate properties includes the contour plots in [1]} and the functional boxplot [4]} of proposed test functions in [5]}.
Formal testing approaches have also been developed for the assessment of these properties. Examples include the test based on spectral representations [6]} and the likelihood ratio test [7]} for the univariate separability property, the separability and symmetry tests from constructed contrasts of convariances [8]}, and the rank-based testing procedure using functional data analysis techniques [5]};
see [10]} for a comprehensive review of univariate spatio-temporal covariance structures and models.
When data are multivariate space-time, there are more types of covariance structures, and the assessment is more complicated.
[11]} used graphical evidence to indicate multivariate separability and symmetry, and [12]} extended their previous testing procedure of the univariate covariance to the multivariate case, using contrasts of covariances at chosen lags.
